Algebra Examples

$\sqrt{x + 3} + \sqrt{x - 3} = 3$

Step 1

Subtract $\sqrt{x - 3}$ from both sides of the equation.

$\sqrt{x + 3} = 3 - \sqrt{x - 3}$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x + 3}}^{2} = {(3 - \sqrt{x - 3})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

${({(x + 3)}^{\frac{1}{2}})}^{2} = {(3 - \sqrt{x - 3})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x + 3)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{x - 3})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(x + 3)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{x - 3})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 3)}^{1} = {(3 - \sqrt{x - 3})}^{2}$

Step 3.2.1.2

Simplify.

$x + 3 = {(3 - \sqrt{x - 3})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(3 - \sqrt{x - 3})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(3 - \sqrt{x - 3})}^{2}$ as $(3 - \sqrt{x - 3}) (3 - \sqrt{x - 3})$ .

$x + 3 = (3 - \sqrt{x - 3}) (3 - \sqrt{x - 3})$

Step 3.3.1.2

Expand $(3 - \sqrt{x - 3}) (3 - \sqrt{x - 3})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x + 3 = 3 (3 - \sqrt{x - 3}) - \sqrt{x - 3} (3 - \sqrt{x - 3})$

Step 3.3.1.2.2

Apply the distributive property.

$x + 3 = 3 \cdot 3 + 3 (- \sqrt{x - 3}) - \sqrt{x - 3} (3 - \sqrt{x - 3})$

Step 3.3.1.2.3

Apply the distributive property.

$x + 3 = 3 \cdot 3 + 3 (- \sqrt{x - 3}) - \sqrt{x - 3} \cdot 3 - \sqrt{x - 3} (- \sqrt{x - 3})$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $3$ by $3$ .

$x + 3 = 9 + 3 (- \sqrt{x - 3}) - \sqrt{x - 3} \cdot 3 - \sqrt{x - 3} (- \sqrt{x - 3})$

Step 3.3.1.3.1.2

Multiply $- 1$ by $3$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - \sqrt{x - 3} \cdot 3 - \sqrt{x - 3} (- \sqrt{x - 3})$

Step 3.3.1.3.1.3

Multiply $3$ by $- 1$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} - \sqrt{x - 3} (- \sqrt{x - 3})$

Step 3.3.1.3.1.4

Multiply $- \sqrt{x - 3} (- \sqrt{x - 3})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + 1 \sqrt{x - 3} \sqrt{x - 3}$

Step 3.3.1.3.1.4.2

Multiply $\sqrt{x - 3}$ by $1$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + \sqrt{x - 3} \sqrt{x - 3}$

Step 3.3.1.3.1.4.3

Raise $\sqrt{x - 3}$ to the power of $1$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + {\sqrt{x - 3}}^{1} \sqrt{x - 3}$

Step 3.3.1.3.1.4.4

Raise $\sqrt{x - 3}$ to the power of $1$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + {\sqrt{x - 3}}^{1} {\sqrt{x - 3}}^{1}$

Step 3.3.1.3.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + {\sqrt{x - 3}}^{1 + 1}$

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + {\sqrt{x - 3}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{x - 3}}^{2}$ as $x - 3$ .

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Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + {({(x - 3)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + {(x - 3)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + {(x - 3)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + {(x - 3)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + {(x - 3)}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$x + 3 = 9 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + x - 3$

Step 3.3.1.3.2

Subtract $3$ from $9$ .

$x + 3 = 6 - 3 \sqrt{x - 3} - 3 \sqrt{x - 3} + x$

Step 3.3.1.3.3

Subtract $3 \sqrt{x - 3}$ from $- 3 \sqrt{x - 3}$ .

$x + 3 = 6 - 6 \sqrt{x - 3} + x$

Step 4

Solve for $- 6 \sqrt{x - 3}$ .

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Step 4.1

Rewrite the equation as $6 - 6 \sqrt{x - 3} + x = x + 3$ .

$6 - 6 \sqrt{x - 3} + x = x + 3$

Step 4.2

Move all terms not containing $- 6 \sqrt{x - 3}$ to the right side of the equation.

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Step 4.2.1

Subtract $6$ from both sides of the equation.

$- 6 \sqrt{x - 3} + x = x + 3 - 6$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$- 6 \sqrt{x - 3} = x + 3 - 6 - x$

Step 4.2.3

Combine the opposite terms in $x + 3 - 6 - x$ .

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Step 4.2.3.1

Subtract $x$ from $x$ .

$- 6 \sqrt{x - 3} = 0 + 3 - 6$

Step 4.2.3.2

Add $0$ and $3$ .

$- 6 \sqrt{x - 3} = 3 - 6$

Step 4.2.4

Subtract $6$ from $3$ .

$- 6 \sqrt{x - 3} = - 3$

Step 5

To remove the radical on the left side of the equation, square both sides of the equation.

${(- 6 \sqrt{x - 3})}^{2} = {(- 3)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

${(- 6 {(x - 3)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 6 {(x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 6 {(x - 3)}^{\frac{1}{2}}$ .

${(- 6)}^{2} {({(x - 3)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.2.1.2

Raise $- 6$ to the power of $2$ .

$36 {({(x - 3)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$36 {(x - 3)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$36 {(x - 3)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$36 {(x - 3)}^{1} = {(- 3)}^{2}$

Step 6.2.1.4

Simplify.

$36 (x - 3) = {(- 3)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$36 x + 36 \cdot - 3 = {(- 3)}^{2}$

Step 6.2.1.6

Multiply $36$ by $- 3$ .

$36 x - 108 = {(- 3)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Raise $- 3$ to the power of $2$ .

$36 x - 108 = 9$

Step 7

Solve for $x$ .

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Step 7.1

Move all terms not containing $x$ to the right side of the equation.

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Step 7.1.1

Add $108$ to both sides of the equation.

$36 x = 9 + 108$

Step 7.1.2

Add $9$ and $108$ .

$36 x = 117$

Step 7.2

Divide each term in $36 x = 117$ by $36$ and simplify.

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Step 7.2.1

Divide each term in $36 x = 117$ by $36$ .

$\frac{36 x}{36} = \frac{117}{36}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $36$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{36 x}{36} = \frac{117}{36}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{117}{36}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Cancel the common factor of $117$ and $36$ .

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Step 7.2.3.1.1

Factor $9$ out of $117$ .

$x = \frac{9 (13)}{36}$

Step 7.2.3.1.2

Cancel the common factors.

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Step 7.2.3.1.2.1

Factor $9$ out of $36$ .

$x = \frac{9 \cdot 13}{9 \cdot 4}$

Step 7.2.3.1.2.2

Cancel the common factor.

$x = \frac{9 \cdot 13}{9 \cdot 4}$

Step 7.2.3.1.2.3

Rewrite the expression.

$x = \frac{13}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13}{4}$

Decimal Form:

$x = 3.25$

Mixed Number Form:

$x = 3 \frac{1}{4}$